We consider groups G such that the set of all values of a fixed word w in G is covered by a finite set of cyclic subgroups. Fernández-Alcober and Shumyatsky studied such groups in the case when w is the word [x1,x2] and proved that in this case the corresponding verbal subgroup G' is either cyclic or finite. Answering a question asked by them, we show that this is far from being the general rule. However, we prove a weaker form of their result in the case when w is either a lower commutator word or a non-commutator word, showing that in the given hypothesis the verbal subgroup w(G) must be finite-by-cyclic. Even this weaker conclusion is not universally valid: it fails for verbose words
For n a positive integer, a group G is called core‐n if H/HG has order at most n for every subgroup H of G (where HG is the normal core of H, the largest normal subgroup of G contained in H). It is proved that a locally finite core‐n group G has an abelian subgroup whose index in G is bounded in terms of n. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.
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